The present invention, in some embodiments thereof, relates to an image processing and, more particularly, but not exclusively, to method and system for processing stochastic and deterministic fields in an image.
One of the measurements that may be used for varying among different visual and/or graphical objects is texture. Texture may be defined as a structure that is made up of a large ensemble of elements that significantly resemble each other, organized according to some kind of ‘order’ in their locations. The elements are organized such that there is not a specific element that attracts the viewer's eyes, but the human viewer gets an impression of uniformity when he looks at the texture, see J. M. Francos, A. Z. Meiri and B. Porat, “A Unified Texture Model Based on a 2-D Wold-like Decomposition”, IEEE Transactions on Signal Processing, Vol. 41, No. 8, August 1993 which is incorporated herein by reference.
The variety of existing textures may be classified in between two extreme categories:
Totally deterministic textures—textures of this type may be described by their primitives (or: cells) together with placement rules that define the exact location of each primitive in the texture. Examples for such a texture are a chessboard and a brick wall.
Purely stochastic textures—textures of this type are efficiently parameterized by random field analysis methods, such as Markov random fields or the auto-regressive model.
Most of the natural textures do not fall into any of the two categories mentioned above. For example, the cellular textures may have primitive cells that are not identical, although they are very similar. Furthermore, the placement rules of the primitive cells need not be totally deterministic, but the viewer will still define the texture as a cellular one. Examples for such textures are a coin collection or scattered coffee beans.
In correspondence with the above mentioned categorization of textures into ‘deterministic’ and ‘stochastic’ types, there are two main approaches to the analysis and synthesis of textures: structural and statistical.
In the structural methods, the texture is considered a cellular and ordered phenomenon. The texture is characterized accordingly by a description of its primitives and their placement rules. The placement rules may be deterministic for periodic and well-structured textures, or stochastic for more random textures, see G. C. Cross and A. K. Jain, “Markov Random Field Texture Models”, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-5, 1983, pp. 25-39, which is incorporated herein by reference. In W. Matusik, M. Zwicker and F. Durand, “Texture Design Using a Simplicial Complex of Morphable Textures”, International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2005, pp. 784-794, which is incorporated herein by reference, Fourier analysis method is applied in order to identify the 2-D fundamental spatial frequency of the texture by locating the spectral density function maxima. This information is used to evaluate the texture placement rule and to isolate the texture primitives. The synthesis procedure places an “averaged” texture cell, according to the found placement rule. The basic disadvantage of this method is its preliminary assumption on the complete periodic structure of the texture field, which results in a synthetic look of the generated textures.
In the statistical methods, the texture is described by a collection of statistics of selected features. Many of these methods rely on the findings in Chandra, M. Petron and R. Piroddi, “Texture Interpolation Using Ordinary Kriging”, Springer, 2005, which is incorporated herein by reference, according to which the human vision system uses global first and second order statistics for texture discrimination. On the basis of this work, an algorithm for synthesizing a texture field from its estimated global second-order statistics is presented in J. Wu, Q. Ruan and G. An, “A Novel Image Interpolation Method Based on Both Local and Global Information”, Springer Berin, Vol. 4681, 2007, which is incorporated herein by reference. The texture is assumed to be a 2-D random field that is described by co-occurrence probabilities, the joint probability of occurrence of two gray levels, conditioned on their relative positions. The co-occurrence probabilities are estimated by the second-order sample mean. The method gives good results for homogeneous purely random texture fields, but is not suitable for more structural textures.
It is important to note that the use of spectral analysis for texture characterization is a natural choice, since the spectral content is strongly related to the spatial variation of the texture. Fine textures have a spectrum rich in high frequencies, while coarse textures are rich in low spatial frequencies.
In L. Wang and K. Mueller, “Generating Sub-Spatial resolution Detail in Images and Volumes Using Constrained Texture Synthesis”, Proceeding of IEEE Conference on Visualization, October 2004, pp. 75-82, which is incorporated herein by reference, the use of a 2-D, auto-regressive, non-causal model for synthesizing textures is suggested. This model characterizes the gray level of a pixel as a linear combination of the gray levels at neighboring pixels and an additive noise.